# Prove the inequality $\sqrt{a^2 + b^2} \geq \frac{|a-b|}{\sqrt{2}}$

Please help me to prove the inequality $$\sqrt{a^2 + b^2} \geq \frac{|a-b|}{\sqrt{2}}.$$

Hint: Square both sides and multiply by $2$, and you'll find that this is equivalent to proving that
$$2(a^2 + b^2) \ge |a - b|^2 = (a - b)^2$$
Hint: Note that $(a-b)^2+(a+b)^2=2(a^2+b^2)$.